Question

Time period depends on Gravitational constant (G),plank constant , (h) and speed of light (c), then T is proportional to

• A. $\frac{G^{1/2}{h}^{1/2}}{C^{3/2}}$
• B. $\frac{G^{1/2}h}{C^{5/2}}$
• C. $\frac{G{h}^{1/2}}{C^{5/2}}$
• D. $\frac{G^{1/2}{h}^{1/2}}{C^{5/2}}$

Answer 1

The correct option is A $\frac{G^{1/2}{h}^{1/2}}{C^{3/2}}$

Dimension of $h=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-1}\right]$

Dimension of $G=\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M^2\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$

Now let, $h^x{C}^y{G}^z$

$\left[L\right]={\left[M{L}^2{T}^{-1}\right]}^x{\left[L{T}^{-1}\right]}^y{\left[M^{-1}{L}^3{T}^{-2}\right]}^z\left[L\right]=\left[M^{x-2}\right]\left[L^{2x+y+3z}\right]\left[T^{-x-y-2z}\right]$

Comparing both sides,

$x-z=0⟹x=z----\left(i\right)2x+y+3z=1⟹y+5z=1----\left(ii\right)-x-y-2z=0⟹x+y+2z=0$

From (i), $y+3z=0----\left(iii\right)$

From (ii) and (iii),

$z=\frac{1}{2},x=\frac{1}{2},y=-\frac{3}{2}$

$\therefore$Dimension of length$=\frac{\sqrt{Gh}}{C^{\frac{3}{2}}}=\frac{G^{\frac{1}{2}}{h}^{\frac{1}{2}}}{C^{\frac{3}{2}}}$